Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily hig...

The immiscible displacement of oil by water through a porous and permeable reservoir rock can be described by the use of a fractional flow curves (fw versus Sw). Water flooding project parameters can be obtained from the fractional flow curve. However, developing a representative fractional flow curve for a specific reservoir can be quite challenging when fluid and special core analysis data is...

in this paper, we develop a numerical resolution of the space-time fractional advection-dispersion equation. we utilize spectral-collocation method combining with a product integration technique in order to discretize the terms involving spatial fractional order derivatives that leads to a simple evaluation of the related terms. by using bernstein polynomial basis, the problem is transformed in...

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the...

‎in the present paper we consider a time-fractional inverse diffusion problem‎, ‎where data is given at $x=1$ and the solution is required in the interval $0

A new method of constructing structure equations of Lie symmetry pseudo-groups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudo-groups, is presented, and illustrated by the examples of the Kadomtsev–Petviashvili and Korteweg–de-Vries equations.

In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula o...

The dynamics of the nonlinear Helmholtz Oscillator with fractional order damping is studied in detail. The discretization of the differential equations according to the Grünwald-Letnikov fractional derivative definition in order to get numerical simulations is reported. Comparison between solutions obtained through a fourth-order Runge-Kutta method and the fractional damping system is commented...